Simple probability distributions on a Fock-space lattice
SSimple probability distributions on a Fock-space lattice
Staszek Welsh and David E. Logan
Department of Chemistry, Physical and Theoretical Chemistry,Oxford University, South Parks Road, Oxford, OX1 3QZ, United Kingdom (Dated: January 10, 2019)We consider some aspects of a standard model employed in studies of many-body localiza-tion: interacting spinless fermions with quenched disorder, for non-zero filling fraction, here on d -dimensional hypercubic lattices. The model may be recast as an equivalent tight-binding modelon a ‘Fock-space (FS) lattice’ with an extensive local connectivity. In the thermodynamic limit exactresults are obtained for the distributions of local FS coordination numbers, FS site-energies, and thedensity of many-body states. All such distributions are well captured by exact diagonalisation onthe modest system sizes amenable to numerics. Care is however required in choosing the appropriatevariance for the eigenvalue distribution, which has implications for reliable identification of mobilityedges. PACS numbers: 71.23.-k, 71.10.-w, 05.30.-d
I. INTRODUCTION
In recent years there has been great interest in thestudy of highly excited quantum states of disordered, in-teracting systems, and notably the phenomenon of many-body localization (MBL); for topical reviews see e.g.[2,3]. In particular it is now well appreciated that ininteracting systems, localization or its absence can beviewed quite generally as an Anderson localization prob-lem in the Fock or Hilbert space of associated many-bodystates. This connection was made long ago in the con-text of the problem of quantum ergodicity in isolatedmolecules, and later in quantum dots. One of the central models extensively studied in MBLis that of interacting spinless fermions with quenched dis-order, for non-zero fermion filling fractions. We considerit here, on d -dimensional hypercubic lattices. As detailedin sec. II, the model can be mapped exactly onto a tight-binding model on a ‘Fock-space lattice’ with an exten-sive local connectivity, each site of which corresponds toa disordered, interacting many-body state with specifiedfermion occupancy of all real-space sites, and is thus as-sociated with a Fock-space ‘site-energy’; with Fock-spacesites connected by the one-electron hopping matrix ele-ment of the original Hamiltonian.While MBL per se is not considered directly in thispaper, we believe it is instructive to have some broadunderstanding of the Fock-space lattice in a statisticalsense. To that end we consider the distributions, overboth Fock-space sites and (where relevant) disorder re-alisations, of the local Fock-space coordination numbers,site-energies, and many-body eigenvalues (or density ofstates). None of these quantities contain informationabout localization itself, but all reflect ‘basic’ propertiesof the system in an obvious sense, and understandingthem is a precursor to considering localization (to whichwe turn in a subsequent work ).As expected from the central limit theorem all suchdistributions are normal, with extensive means; the rel-evant moments are obtained (sec. III) as a function of the bare model parameters, fermion filling fraction andspace dimension. A subtlety arising in the choice of vari-ance for the eigenvalue spectrum (and Fock-space site-energies) is pointed out in sec. IV, with the correct choiceenabling a pristine distinction between localized and ex-tended states as a function of energy, as required for areliable identification of mobility edges which may sep-arate them. Comparison to exact diagonalisation showsthe distributions considered to be well captured even forthe small system sizes practically accessible to numericsin d = 1 ,
2. Concluding remarks are given in sec. V.
II. MODEL
The Hamiltonian is a standard model of spinlessfermions, here considered on a d -dimensional lattice: H = H W + H t + H V (1a)= (cid:88) i (cid:15) i ˆ n i + (cid:88) (cid:104) ij (cid:105) t ( c † i c j + c † j c i ) + (cid:88) (cid:104) ij (cid:105) V ˆ n i ˆ n j (1b)with ˆ n i = c † i c i . The hoppings ( t ) and interactions ( V )are nearest neighbour (NN), with (cid:104) ij (cid:105) denoting distinctNN pairs. The site energies { (cid:15) i } in H W are characterisedby quenched random disorder, with a distribution P ( (cid:15) i )common to all sites (and chosen for convenience to havezero mean, (cid:104) (cid:15) (cid:105) = (cid:82) d(cid:15)P ( (cid:15) ) (cid:15) = 0). The eigenvalues of H are denoted by E n .The lattice has N sites and contains N e fermions. Weare interested in the thermodynamic limit where both N ≡ L d and N e → ∞ , holding the filling ν = N e /N fixed and non-vanishing. The coordination number ofthe lattice is denoted by Z d , with Z d = 2 d for hypercubiclattices. a r X i v : . [ c ond - m a t . s t r- e l ] J a n l = 0 l = 1 l = 2 l = 3...... l = 35 l = 36 | i | i| i | i | i| i | i FIG. 1. Example of Fock-space lattice (for an open 1 d chain with N = 12 sites and N e = 6 fermions, N H = 924). Each siterepresents a state | I (cid:105) , solid lines show coupling between states under hopping. See text for discussion. Inset : a small sectionshowing 7 sites, with coordination numbers Z I ranging from 5 − A. Equivalent tight-binding model
For any given fermion number N e , the dimension ofthe associated Fock space (FS) is N H = N C N e ( ≡ (cid:0) NN e (cid:1) ),growing exponentially with the number of sites, N H ∝ N − / e cN (with c = − [ ν ln ν + (1 − ν ) ln(1 − ν )] the con-figurational entropy per site).The Hamiltonian may be recast as an effective tight-binding model (TBM) on a lattice of N H ‘sites’. To thisend, first separate H = H + H t with H = H W + H V (eq. 1). Since H involves solely number operators, itsstates {| I (cid:105)} are | I (cid:105) = |{ n ( I ) i }(cid:105) = ( c † ) n ( I )1 ( c † ) n ( I )2 .... ( c † N ) n ( I ) N | vac (cid:105) (2)with each occupation number n ( I ) i = 0 or 1 only, and (cid:80) i n ( I ) i = N e for any | I (cid:105) . These states are orthonormal,with energies E I under H given by E I = (cid:88) i (cid:15) i n ( I ) i + V r I , r I = (cid:88) (cid:104) ij (cid:105) n ( I ) i n ( I ) j (3)where r I is thus defined; and each of the N H configu-rations of fermions on the lattice specifies uniquely onesuch basis state | I (cid:105) .The {| I (cid:105)} may be viewed as forming a lattice of N H sites in state-space, here referred to as a ‘FS lattice’.These sites are connected under the hopping term H t (eq. 1), with T IJ = (cid:104) I | H t | J (cid:105) = T JI the real symmetricmatrix elements coupling them. Since the hopping is NNonly, | T IJ | = t for T IJ non-vanishing ( T IJ is either ± t for general d ≥
2, its sign obviously depending on the configuration of fermions in | I (cid:105) and | J (cid:105) ; while T IJ = t for all connected FS sites in the d = 1 open chain). Thetotal Hamiltonian can thus be written as H = (cid:88) I E I | I (cid:105)(cid:104) I | + (cid:88) I,J ( J (cid:54) = I ) T IJ | I (cid:105)(cid:104) J | . (4)This has the form of a TBM on the FS lattice; with thenumber of sites to which any given site I is connectedunder the hopping defining the local coordination numberof the site, denoted Z I .Given the mapping to an effective TBM, the samequestions can obviously be asked about this FS latticeas arise for a one-electron disordered TBM. What forexample is the disorder-averaged density of many-bodystates; and are those states of some given energy local-ized on a vanishingly small fraction of the FS lattice, ordelocalized over a finite fraction of it? The mapping alsomeans that some techniques applicable to one-electronproblems on a real-space lattice may be extended to en-compass the question of MBL in the FS lattice (as wewill consider in subsequent work ). B. Fock space lattice
We have found it helpful to have a concrete pictureof the FS lattice, in part to prompt questions about itsbasic characteristics and their implications. An illustra-tive example is given in fig. 1, shown for a small 1 d openchain of N = 12 sites, N e = 6 fermions (filling ν = 1 / | I (cid:105) ), and solid lines denote connections between the statesunder the hopping. The sites are arranged in rows, withrow index l . The top row ( l = 0) consists of the singlestate | I (cid:105) = | L (cid:105) = | (cid:105) in which all fermionsoccupy the leftmost real-space sites. This state has acoordination number Z I = 1, since under H t it can con-nect only to the single state | (cid:105) in row l = 1.The latter connects under H t to two further states in row l = 2 (such that its coordination number Z I = 3); andthese two states are connected in turn to a total of threestates in row l = 3. The process continues in this fash-ion, with the number of sites/states in successive layersfirst growing and then decreasing as seen in fig. 1 (re-flecting that a fermion cannot move beyond the end sitefor an open chain); and terminates at the bottom state | I (cid:105) = | R (cid:105) = | (cid:105) in which all fermions occupythe rightmost real-space sites (row l = 36 here).Generally, the total number of rows in the FS lattice is1 + N e ( N − N e ) = 1 + ν (1 − ν ) N (as the minimum num-ber of sequential hops required to connect | L (cid:105) to | R (cid:105) is N e ( N − N e )). Since the FS lattice is invariant to readingthe fermion strings in FS sites from left to right, or viceversa, the number of FS sites in rows l and N e ( N − N e ) − l are the same. The number of sites in row l can be shownto be the number of integer partitions of l , subject tothe restriction that no partition may have more than N e parts and no individual part can be larger than N − N e ;and asymptotically grows exponentially in √ l as l is in-creased towards the middle rows l ∼ ν (1 − ν ) N .Although our focus here is on finite filling ν = N e /N inthe thermodynamic limit N → ∞ , for which the FS lat-tice clearly ‘balloons’, we add that in the single-fermionlimit N e = 1 it reduces simply to that for the real-spacelattice; with N rows, each containing just one state (inwhich the fermion occupies a particular real-space site).Several general features of the FS lattice are appar-ent. First, the hopping T IJ is local in state-space, in thesense that the FS sites {| J (cid:105)} to which any given | I (cid:105) con-nects under H t necessarily lie in a directly adjacent row,reflecting that one fermion has hopped to a NN site inthe physical lattice. Moreover, since the hypercubic real-space lattices considered are bipartite, it follows that thecorresponding FS lattice is also bipartite.Second, physical properties of the system are typicallycharacterised by probability distributions – over FS lat-tice sites, disorder realisations, or both. An obvious ex-ample is the local connectivity under T IJ , reflected in thedistribution of coordination numbers Z I over FS sites(which is independent of disorder). In e.g. a 1 d chainthe minimum Z I for all fillings ν = N e /N is clearly Z I, min = 1 (or 2 for periodic boundary conditions), oc-curring solely for the apical sites of rows l = 0 and N e ( N − N e ) where the fermions are maximally bunched.The maximum Z I by contrast is macroscopically large, Z I, max = 2 N e for all fillings ν < /
2, arising for statesin which each fermion is surrounded by at least twoempty sites; with Z I, max = 2( N − N e ) for ν > / Z I, max = N − {E I } are the counterparts, for the equivalent Fock-spaceTBM eq. 4, of the site-energies { (cid:15) i } in a one-particleTBM. One thus expects their distribution to influencewhether many-body states are FS localized or extended;while recognising that, unlike the ( N ) real-space { (cid:15) i } which are independent random variables, the N H FS site-energies {E I } are correlated. A further obvious exampleis the eigenvalue spectrum of H – or density of states(DoS) – and its distribution over disorder realisations. III. DISTRIBUTIONS
We seek then the distributions, over FS lattice sites anddisorder realisations, of: the FS site-energies {E I } , theeigenvalues of H , the coordination numbers Z I , and r I = (cid:80) (cid:104) ij (cid:105) n ( I ) i n ( I ) j of eq. 3 (which determines the interactioncontribution to E I ).In the thermodynamic limit of interest we take it asgiven that all such distributions are Gaussian (as is es-sentially obvious from the central limit theorem, althoughcan be shown explicitly). We thus focus on first and sec-ond moments. As will be seen, all such may be obtainedsolely from a knowledge of that for r I .For any quantity O I = (cid:104) I | ˆ O | I (cid:105) , its average O over bothFS sites and disorder realisations is O = (cid:104) Tr O I (cid:105) (cid:15) . (5)Here (cid:104) .... (cid:105) (cid:15) = (cid:82) (cid:81) Ni =1 [ d(cid:15) i P ( (cid:15) i )] .... denotes the disorderaverage, and Tr O I an average over the N H = N C N e FSsites/statesTr O I = N − H (cid:88) I O I = N − H (cid:88) I (cid:104) I | ˆ O | I (cid:105) (6)(such that averages over disorder and FS sites commute, (cid:104) Tr O I (cid:105) (cid:15) ≡ Tr (cid:104) O I (cid:105) (cid:15) ).Consider a generic product ( n ( I ) i n ( I ) j n ( I ) k ..... ) corre-sponding to m distinct real-space sites i, j, k... . This isnon-zero only if all m sites are occupied (occupation num-bers of 1), so (cid:80) I ( n ( I ) i n ( I ) j n ( I ) k ..... ) is simply the number ofways of distributing ( N e − m ) fermions over ( N − m ) sites.Hence Tr( n ( I ) i n ( I ) j n ( I ) k ... ) = N − H × ( N − m ) C ( N e − m ) ≡ ν m isTr( n ( I ) i n ( I ) j n ( I ) k ..... ) = ν m = m − (cid:89) n =0 N e − nN − n = ν m − (cid:89) n =1 ν − nN − nN (7)such that ν = ν , the filling fraction. Note that ν m ≡ ν m + O (1 /N ) in the thermodynamic limit (although eq.7 holds for any N, N e , and is required below).Consider first the FS site-energies E I = (cid:104) I | H | I (cid:105) (eq.3). Tr E I is the mean of E I over FS sites for a givendisorder realisation, given (via eq. 7) byTr E I = ν (cid:88) i (cid:15) i + V r, (8)with r ≡ Tr r I (as r I is independent of disorder). Sincethe disorder-averaged (cid:104) (cid:15) (cid:105) (cid:15) = 0, E = (cid:104) Tr E I (cid:105) (cid:15) = V r . Fromeqs. 3,7 Tr r I = ν (cid:80) (cid:104) ij (cid:105) = ν Z d N/
2, so E is given in thethermodynamic limit by E = V r = V Z d ν N = V νdN e . (9)Disorder-induced fluctuations in Tr E I are embodied in (cid:104) [Tr E I ] (cid:105) (cid:15) , and since the lattice site-energies { (cid:15) i } areindependent random variables, (cid:104) (cid:15) i (cid:15) j (cid:105) (cid:15) = δ ij (cid:104) (cid:15) (cid:105) (with (cid:104) (cid:15) (cid:105) ≡ (cid:104) (cid:15) (cid:105) (cid:15) for brevity); so (cid:104) [Tr E I ] (cid:105) (cid:15) = ν (cid:104) (cid:15) (cid:105) N + V r . (10) E = (cid:104) Tr( E I ) (cid:105) (cid:15) likewise follows using eqs. 3,7 as (cid:104) Tr E I (cid:105) (cid:15) = E = ν (cid:104) (cid:15) (cid:105) N + V r (11)(where r = Tr r I ). For reasons explained in sec. IVwe now define two distinct variances, specifically µ E = (cid:104) Tr([ E I − Tr E I ] ) (cid:105) (cid:15) and µ E = (cid:104) Tr([ E I − (cid:104) Tr E I (cid:105) (cid:15) ] ) (cid:105) (cid:15) = (cid:104) Tr([ E I − E ] ) (cid:105) (cid:15) . These are thus given by µ E = (cid:104) Tr E I (cid:105) (cid:15) − (cid:104) [Tr E I ] (cid:105) (cid:15) = ν (1 − ν ) (cid:104) (cid:15) (cid:105) N + V µ r (12a) µ E = (cid:104) Tr E I (cid:105) (cid:15) − (cid:104) Tr E I (cid:105) (cid:15) = ν (cid:104) (cid:15) (cid:105) N + V µ r (12b)with µ r = r − r the variance of r I .We turn now to the distribution of eigenvalues. Tr H does not depend on the basis chosen, so Tr H = N − H (cid:80) I (cid:104) I | H | I (cid:105)≡ N − H (cid:80) n E n , and gives the centre ofgravity of the eigenvalues { E n } for any given disorderrealisation. Since Tr H = Tr E I (eq. 4), the disorder-averaged mean eigenvalue E = (cid:104) Tr H (cid:105) (cid:15) = E , i.e. E = E = V r = V νdN e (13)(with E ∝ N e reflecting extensivity). Since (cid:104) [Tr H ] (cid:105) (cid:15) = (cid:104) [Tr E I ] (cid:105) (cid:15) , eq. 10 gives (cid:104) (cid:2) Tr H (cid:3) (cid:105) (cid:15) = ν (cid:104) (cid:15) (cid:105) N + V r . (14)Now consider E = (cid:104) Tr H (cid:105) (cid:15) . From eq. 4, (cid:104) I | H | I (cid:105) = E I + (cid:80) J T IJ T JI = E I + t Z I with Z I the coordinationnumber of FS site I . Hence (cid:104) Tr H (cid:105) (cid:15) = E + t Z (15)with Z = Tr Z I the average coordination number of theFS lattice.In parallel to the FS site energies above, we defineagain two variances, µ E = (cid:104) Tr([ H − Tr H ] ) (cid:105) (cid:15) and µ E = (cid:104) Tr([ H − (cid:104) Tr H (cid:105) (cid:15) ] ) (cid:105) (cid:15) = (cid:104) Tr([ H − E ] ) (cid:105) (cid:15) ; which follow as µ E = (cid:104) Tr (cid:0) [ H − Tr H ] (cid:1) (cid:105) (cid:15) = ν (1 − ν ) (cid:104) (cid:15) (cid:105) N + V µ r + t Z (16a) µ E = (cid:104) Tr (cid:0) [ H − (cid:104) Tr H (cid:105) (cid:15) ] (cid:1) (cid:105) (cid:15) = ν (cid:104) (cid:15) (cid:105) N + V µ r + t Z. (16b) Physically, µ E gives the disorder-averaged variance of theeigenvalues, relative to their centre of gravity Tr H = N − H (cid:80) n E n for each disorder realisation; while µ E givesthe variance relative to the full mean (cid:104) Tr H (cid:105) (cid:15) over bothFS sites and disorder. The reasons for introducing thesetwo distinct variances are discussed in sec. IV. Here wesimply note that it is µ E (and likewise µ E ) which is ofprimary relevance.The variances in eqs. 12,16 thus follow once µ r and themean FS coordination number Z are known. Obviouslyneither of the latter depends on disorder. Further, as nowshown, the coordination number Z I is simply related to r I , so only the latter need be considered.Each configuration of fermions on the real-space latticespecifies uniquely one FS basis state | I (cid:105) . With 1 denotingan occupied site and 0 an empty site, the following typesof NN pairs arise: 11, 10 and 00. The total number ofsuch pairs in any | I (cid:105) are denoted N ( I ) mn , and are clearly N ( I ) = (cid:88) (cid:104) ij (cid:105) n ( I ) i n ( I ) j = r I (17a) N ( I ) = (cid:88) (cid:104) ij (cid:105) (cid:2) n ( I ) i (1 − n ( I ) j ) + (1 − n ( I ) i ) n ( I ) j (cid:3) = Z I (17b) N ( I ) = (cid:88) (cid:104) ij (cid:105) (1 − n ( I ) i )(1 − n ( I ) j ) . (17c)Using (cid:80) (cid:104) ij (cid:105) = Z d N/
2, eqs. 17 naturally sum to the totalnumber of NN pairs, N ( I ) + N ( I ) + N ( I ) = N Z d = dN .Obviously N ( I ) = r I (see eq. 3). Equally obviously N ( I ) = Z I , the coordination number of FS site I , be-cause each 10-pair enables a fermion to hop under t ; andusing (cid:80) (cid:104) ij (cid:105) n ( I ) i = Z d (cid:80) i n ( I ) i = Z d N e , eq. 17b gives Z I = Z d N e − r I = 2( dN e − r I ) . (18)The distribution of FS coordination numbers thus followsdirectly from a knowledge of that for r I . Eq. 18 is alsophysically clear: for r I = N ( I ) = 0, all NN sites to eachoccupied site in | I (cid:105) are empty, so each of the N e fermionscan hop under t to its 2 d NNs, whence Z I = 2 dN e ; whilefor r I = N ( I ) (cid:54) = 0, each additional NN 11-pair clearly‘blocks’ one hop for each of the pair of fermions, andthus reduces Z I by 2. A. Variance of r I From eqs. 12,16 the final step is to determine the sec-ond moment r = Tr (cid:80) (cid:104) ij (cid:105) (cid:80) (cid:104) kl (cid:105) n ( I ) i n ( I ) j n ( I ) k n ( I ) l . Sincethe sites in (cid:104) ij (cid:105) and (cid:104) kl (cid:105) are not all distinct we partitionthis sum into terms involving 2 NN sites, 3 adjacent NNsites, and two pairs of distinct NN sites, specifically r = (cid:88) (cid:104) ij (cid:105) ν + 2 (cid:88) (cid:104) ij (cid:105) (cid:88) k ν + (cid:88) (cid:104) ij (cid:105) , (cid:104) kl (cid:105)(cid:48) ν . (19)The middle sum refers to 3 distinct sites, where the site k (cid:54) = i or j , but is a NN to one of them, such that for anygiven (cid:104) ij (cid:105) there are Z d − k sum. Hence,2 (cid:80) (cid:104) ij (cid:105) (cid:80) k ν = 2( Z d − ν (cid:80) (cid:104) ij (cid:105) = N Z d ( Z d − ν .The final sum in eq. 19 refers to two pairs of distinct NNsites. Recognising that the original sum for r containeda total of [ Z d N ] terms, while the first pair of terms onthe right side of eq. 19 contain respectively Z d N and N Z d ( Z d −
1) terms, gives (cid:80) (cid:48)(cid:104) ij (cid:105) , (cid:104) kl (cid:105) ν = Z d N (cid:2) Z d N − (2 Z d − (cid:3) ν . Using the precise forms for ν m (eq. 7),together with r = Z d N ν , then gives µ r = r − r as µ r = Z d ν (1 − ν ) N (cid:0) N − [ Z d + 1] (cid:1) ( N e − N − N e − N − ( N − N − . (20)This is exact under periodic boundary conditions for fi-nite N . In the thermodynamic limit of interest it givesthe desired result for the variance of r I , µ r = ν (1 − ν ) Z d N = ν (1 − ν ) dN e (21)(with leading corrections O (1)).Let us simply recap the essential results arising in thethermodynamic limit. Eq. 13 gives the mean energies, E = E = V r = V νdN e . For the variances, eqs. 12,16give µ E = µ E + t Z (22a) µ E = µ W + V µ r : µ W = (1 − ν ) (cid:104) (cid:15) (cid:105) N e (22b)and likewise µ E = µ E + t Z (23a) µ E = µ W + V µ r : µ W = (cid:104) (cid:15) (cid:105) N e , (23b)all being a sum of independent contributions from disor-der, interactions and (for the eigenvalues) hopping. Themean FS coordination number follows directly from eq.18 as Z = Z d N e − r , so from eq. 9 for rZ = Z d (1 − ν ) N e = 2 ν (1 − ν ) dN ; (24)while the variance µ Z = 4 µ r (from eq. 18), with µ r given by eq. 21. All these quantities follow directly onspecifying the filling fraction ν = N e /N , the space di-mension d of the real-space lattice, and the number offermions N e . Note also that µ E and µ E are invariantunder a particle-hole transformation ν ↔ − ν (since µ W = ν (1 − ν ) (cid:104) (cid:15) (cid:105) N ); µ E and µ E by contrast are not.The distribution of FS coordination numbers (or equiva-lently r I , eq. 18) itself depends solely on the lattice, andnot on H or its parameters. For the 1 d chain, this distri-bution can be determined for arbitrary N, N e from basiccombinatorics. With p ( N, N e ; Z ) denoting the fraction ofFS lattice sites with coordination number Z , it is givenby (explicitly here for an open chain): N H p ( N, N e ; Z )= 2 ( N − Z ) Z ( N e − C Z − N − N e − C Z − : Z even= 2 ( N e − C Z − ( N − N e − C Z − : Z odd 0 2 4 6 8 10 12 14 1600 . . . . Z
25 50 7500 . . Z FIG. 2. Distribution p ( N, N e ; Z ) of coordination num-bers Z for the FS lattice arising for the 1 d chain, shown for N = 16 , N e = 8; and compared to the normal distribution(solid line) of mean Z (eq. 24) and variance µ Z = 4 µ r (eq.21) appropriate to the thermodynamic limit. Inset : samecomparison for N = 100 , N e = 50. As an indication of the size-dependence of this distri-bution, Fig. 2 shows p ( N, N e ; Z ) vs Z for a half-filled N = 16-site system (as typically employed in exact di-agonalisation studies of MBL, and with N H = 12870FS sites). The Gaussian distribution appropriate to thethermodynamic limit is shown for comparison. Even for N = 16 this is quite well approached, despite the modestrange of available Z ’s; and by N = 100 (fig. 2 inset) thediscrete and normal distributions are barely distinguish-able. Analogous comparison for the eigenvalue distribu-tion is given below. IV. ENERGY VARIANCES
We have introduced two distinct variances for theeigenvalues (eqs. 16), and the FS site-energies (eq. 12).Here we explain why, and why it is that µ E and µ E are thephysically relevant variances; focusing in the following onthe eigenvalues (the same considerations apply to the FSsite-energies). The reason is a little subtle, does not usu-ally arise in considering one-body localization (1BL) –although it may do, as explained – and has implicationsfor the identification of mobility edges.Consider first the 1BL case of a single fermion, withfilling ν = 1 /N . Here, trivially, Z ≡ Z d is the coordina-tion number of the physical lattice and r = 0 = µ r (since N e = 1). In this case (eq. 16), µ E ≡ µ E (= (cid:104) (cid:15) (cid:105) + t Z d )coincide in the thermodynamic limit, so which of the twoone considers is immaterial. This reflects the fact (eq. 14)that for 1BL (cid:104) [Tr H ] (cid:105) (cid:15) = (cid:104) (cid:15) (cid:105) /N vanishes as N → ∞ , i.e.that Tr H = N − H (cid:80) n E n – the centre of gravity of theeigenvalue distribution for any given disorder realisation– does not fluctuate from realisation to realisation. For amacroscopic system, each realisation will then yield thesame eigenvalue spectrum (self-averaging). By itself, thatspectrum does not of course contain information aboutwhether states are localized (L) or extended (E). Butstates of any given energy are either L or E with proba-bility unity over an ensemble of disorder realisations; sothe fact that the same spectrum is obtained for all dis-order realisations ensures a pristine distinction betweenL and E states as a function of energy, and hence anunambiguous identification of mobility edges separatingthem.The situation above is not however ubiquitous, evenfor 1BL. To illustrate this, consider the Hamiltonian H (cid:48) = (cid:88) i (cid:15) i (ˆ n i − ) + H t + (cid:88) (cid:104) ij (cid:105) V (ˆ n i − )(ˆ n j − )(25a)= c + (cid:88) i (cid:15) i (ˆ n i − ) + H t + (cid:88) (cid:104) ij (cid:105) V ˆ n i ˆ n j (25b)(with c = V Z d ( − ν ) N a constant which is irrelevantin the following). As for the H of eq. 1, this Hamiltonianis widely studied in MBL, since in d = 1 it maps directlyto a random XXZ model under a Jordan-Wigner trans-formation. H (cid:48) is regarded as being equivalent to H ; andindeed, aside from the trivial disorder-independent con-stant ( c ), the two Hamiltonians differ only by a constant: H (cid:48) ≡ H − C with C = (cid:80) i (cid:15) i . However C depends onthe disorder realisation, C ≡ C ( { (cid:15) i } ); with a vanishingdisorder-averaged mean C = 0, but a non-zero variance C = (cid:104) (cid:15) (cid:105) N proportional to system size N . For H (cid:48) onehas Tr H (cid:48) = ( ν − ) (cid:80) i (cid:15) i + V r = ( ν − ) C + V r , withthe variances µ E and µ E given by µ E = ν (1 − ν ) (cid:104) (cid:15) (cid:105) N + V µ r + t Z (26a) µ E = (cid:104) (cid:15) (cid:105) N + V µ r + t Z. (26b)Note that µ E = (cid:104) Tr([ H (cid:48) − Tr H (cid:48) ] ) (cid:105) (cid:15) for H (cid:48) is identicalto that arising for H (eq. 16a), so is unaffected by C ≡ C ( { (cid:15) i } ); by contrast, µ E = (cid:104) Tr([ H (cid:48) −(cid:104) Tr H (cid:48) (cid:105) (cid:15) ] ) (cid:105) (cid:15) differsfrom that for H (eq. 16b).We return to this below, but first consider again the1BL limit, where ν = 1 /N . In this case, unlike that for H considered above, µ E and µ E (eqs. 26) no longer co-incide: µ E = (cid:104) (cid:15) (cid:105) + t Z d (as arises also for H ), while µ E = (cid:104) (cid:15) (cid:105) N + t Z d grows with system size N . Thisreflects the fact that the centre of gravity of the eigen-value distribution (Tr H (cid:48) ) fluctuates from realisation torealisation, (cid:104) [Tr H (cid:48) ] (cid:105) (cid:15) = (cid:104) (cid:15) (cid:105) N = C growing with N (in contrast to (cid:104) [Tr H ] (cid:105) (cid:15) = (cid:104) (cid:15) (cid:105) /N ). The implications ofthis are clear – for a macroscopic system each disorderrealisation no longer yields the same eigenvalue spectrumas a function of energy. Rather, any two disorder real-isations will yield identical copies of the spectrum, butenergetically displaced/offset from each other by the dif-ference in the C ≡ C ( { (cid:15) i } )’s for the two realisations.This merely reflects the fact that for any given disorder − − −
10 0 10 20 30 40 5000 . . ω/t D ( ω ) . . d = 21 FIG. 3. Many-body eigenvalue spectrum D ( ω ) vs ω/t (red),with site-energy distribution P ( (cid:15) ) = W θ ( W − | (cid:15) | ), shownfor W/t = 12 , V /t = 2 at half-filling ν = ; obtained nu-merically for an N = 16-site chain (200 disorder realisations),and compared to the Gaussian thermodynamic limit resulteq. 27 with variance µ E of eq. 16a (black line). The corre-sponding ‘fully-averaged’ spectrum of variance µ E discussedin text is also shown (blue). Inset : D ( ω ) vs ω/t (red) foran N = 4 × d = 1 case), again compared to the Gaussian eq. 27 withvariance µ E (black line). realisation, H (cid:48) ≡ H − C and H have exactly the sameeigen states , which hence have the same L or E character;but their eigen values , while in 1:1 correspondence, areeach shifted by the disorder-dependent C ( { (cid:15) i } ). The cureis obvious: to restore the sharp distinction between L andE states as a function of energy, as required for unam-biguous identification of mobility edges, one needs onlyto eliminate these realisation-dependent offsets; therebyreferring all energies to a common origin independentof disorder (specifically (cid:104) Tr H (cid:48) (cid:105) (cid:15) = 0 = (cid:104) Tr H (cid:105) (cid:15) ), witheigenvalue fluctuations treated relative to their centre ofgravity for each disorder realisation. It is of course pre-cisely this which is captured by µ E = (cid:104) Tr([ H (cid:48) − Tr H (cid:48) ] ) (cid:105) (cid:15) (= (cid:104) Tr([ H − Tr H ] ) (cid:105) (cid:15) ), hence our focus on it rather thanon µ E .The 1BL situation just described for the case of H (cid:48) is in fact the norm when considering MBL (where thefilling ν = N e /N is strictly non-vanishing in the thermo-dynamic limit). Here, whether H or H (cid:48) is considered, thecentre of gravity of the eigenvalue distribution fluctuateswith disorder realisation, with (cid:104) [Tr H ] (cid:105) (cid:15) = ν (cid:104) (cid:15) (cid:105) N + V r (eq. 14) and (cid:104) [Tr H (cid:48) ] (cid:105) (cid:15) = [ ν − ] (cid:104) (cid:15) (cid:105) N + V r each inevitably ∝ N . The resolution of the disorder-induced offsets is precisely the same as for the 1BL ex-ample above: all energies are referred to a common origin( E = V r = (cid:104) Tr H (cid:48) (cid:105) (cid:15) = (cid:104) Tr H (cid:105) (cid:15) ), with eigenvalue fluctua-tions treated relative to their centre of gravity (Tr H orTr H (cid:48) ) for each disorder realisation, as embodied in µ E .With this, the (normalised) eigenvalue spectrum D ( ω ) = N − H (cid:80) n δ ( ω − E n ) ( ≡ N − H (cid:104) (cid:80) n δ ( ω − E n ) (cid:105) (cid:15) )is given by the Gaussian D ( ω ) = 1 √ πµ E exp (cid:16) − [ ω − E ] µ E (cid:17) . (27)Note again that µ E is the same for both Hamiltonians H and H (cid:48) (eqs. 16a,26a), whence so too are their spec-tra D ( ω ). Indeed this is readily seen to be true for anyHamiltonian with a site-energy term (cid:80) i (cid:15) i (ˆ n i − ζ ) with ζ an arbitrary disorder-independent constant, encompass-ing H and H (cid:48) as particular cases. It is eq. 27 we refer toin the following as the eigenvalue spectrum/DoS.The previous considerations are salutary. If one doesnot account for disorder-induced energy offsets as above,and instead averages the eigenvalue distribution willynilly over all disorder realisations, then if a sharp dis-tinction between L or E states as a function of energyoccurs, it will be lost (as above). The resultant av-eraged distribution in this case is again Gaussian withthe same mean E = V r , but now with a variance µ E = (cid:104) Tr([ H − (cid:104) Tr H (cid:105) (cid:15) ] ) (cid:105) (cid:15) (with µ E differing for H and H (cid:48) , eqs. 16b,26b respectively, and µ E (cid:54) = µ E for ei-ther H or H (cid:48) ).These differences are clearly evident in finite-size cal-culations. We illustrate them in fig. 3, considering theHamiltonian H (eq. 1) for the d = 1 chain, with a stan-dard box distribution P ( (cid:15) ) = W θ ( W − | (cid:15) | ) for the site-energy distribution (and choosing W/t = 12,
V /t = 2).Results are shown for N = 16 sites at half-filling ν = ,generated from 200 disorder realisations. The resul-tant eigenvalue spectrum is shown [red], and compared(solid line) to the Gaussian D ( ω ) eq. 27 with variance µ E (eq. 16a). The latter is seen to be excellently cap-tured, even for N = 16. The fully-averaged spectrum isalso shown [blue], and likewise compared to a Gaussian ofvariance µ E (eq. 16b), which similarly captures it well.The two spectra are visibly distinct (with µ E > µ E ), asexpected from the considerations above.The inset to fig. 3 also shows the numerical DoS for a d = 2-dimensional square lattice (dimensionality d enter-ing both the mean eigenvalue E , eq. 13, and the varianceeq. 16a via the interaction and hopping terms, V µ r and t Z ). Parameters considered are otherwise the same asthose for d = 1 in the main figure, and comparison ismade to the Gaussian D ( ω ) eq. 27 with variance eq. 16a.Despite the relatively meagre N = 4 × N -dependent, in two ways: via E ∝ N (which is triv-ially dealt with by referring energies relative to the bandcentre ω = E ), and because its standard deviation µ E ∝ √ N . As such, it is natural to rescale energiesas ˜ ω = ( ω − E ) /µ E , such that the DoS ˜ D (˜ ω ) (normalisedto unity over ˜ ω ) is a standard normal distribution,˜ D (˜ ω ) = 1 √ π exp (cid:0) − ˜ ω (cid:1) : ˜ ω = ( ω − E ) µ E . (28) V. CONCLUSION
We have considered a canonical model employed instudies of MBL: a disordered system of interacting spin-less fermions, here on a d -dimensional lattice with N ≡ L d sites and N e fermions, for non-vanishing filling ν = N e /N . The model can be cast as an equivalent tight-binding model (eq. 4) on a locally connected Fock-spacelattice of dimension N H ∝ e cN , the sites of which cor-respond to the many-particle states of the system in theabsence of hopping. As such, precisely the same ques-tions may be asked as for a conventional one-body TBM,including about the density of many-body states, andwhether those states are FS localized or extended. Wehave barely touched on the latter question here; but, as aprecursor to it, have considered the distributions – overFS lattice sites and/or disorder realisations as appropri-ate – of the system’s eigenvalues, the FS site energies, andthe local FS coordination numbers. All such are normallydistributed, with variances in the thermodynamic limitthat are readily determined, and found to be well cap-tured by exact diagonalisation on the small system sizesof up to N = 16 sites typically used in numerical work.Some aspects of the results above warrant final briefcomment. First, from the discussion in sec. IV of theeigenvalue spectrum D ( ω ), eq. 27 (or ˜ D (˜ ω ), eq. 28), allbut an exponentially small fraction of states lie in a ‘ √ N scaling window’ about the band centre ω = E , i.e. onenergy scales set by µ E ∝ √ N . It is worth consideringwhat implications this might have for many-body mobil-ity edges (ME). Above a certain non-zero critical disorder W = W c , states at the band centre – and by presump-tion all states – are MBL. What then happens as W is reduced below W c ? Without prejudice there wouldseem to be two distinct possibilities: either essentiallyall states become extended as W is decreased just below W c ; or not. Were the former to occur, we have nothing tosay about it. But if the latter arises, one expects many-body MEs at [ ω − E ] = ω mob ± to open up continuously about the band centre, separating many-body localizedfrom extended states; as indeed detailed numerical workfinds. In this case we simply point out that – byvirtue of the µ E ∝ √ N scaling of the eigenvalue spec-trum – mobility edge trajectories (as MEs move furtherinto the band with decreasing W ) must likewise scalewith √ N , and not with N . While this does not precludea subsequent crossover to MEs scaling with N itself, bydefinition the latter can occur only when ω mob lies O ( N )away from the band centre; and as such lies deep in thetails of the eigenvalue spectrum, where the fraction ofstates is exponentially small.These considerations have implications for numericallydetermined mobility edges. The energy axis in suchstudies is commonly expressed as an energy density ω = ( ω − E min ) / ( E max − E min ); with E min / max thesmallest/largest eigenvalue, and ω = 1 / E min and E max are howevereach O ( N ) removed in energy from the band centre,with E max − E min ∝ N . Hence if a mobility edge at ω lies within, say, a few µ E ∝ √ N of the band centre,such that a fraction O (1) of states are delocalised, then ω → + O (1 / √ N ) nevertheless ‘sticks’ at 1 / / √ N scaling window. If by con-trast the energy axis is scaled in terms of µ E ∝ √ N , thenthe continuous evolution of mobility edges as they movethrough the eigenvalue spectrum will be captured. Ofcourse these considerations refer to the thermodynamiclimit, and it is not a priori clear whether the modestsystem sizes amenable to numerics would be sufficient inpractice to distinguish between √ N and N behaviour;although re-evaluation of previously obtained numericaldata along the lines suggested here should shed light onthe matter.A further aspect of ω = ( ω − E min ) / ( E max − E min ) –viz. that the band centre is identified as ω = 1 /
2, i.e.by ω = ( E min + E max ) – relates to the discussion ofsec. IV. Since E min / max are by definition the extremaleigenvalues, they lie deep in the exponential tails of theeigenvalue spectrum, and will each fluctuate consider-ably from disorder realisation to realisation. Identifyingthe band centre by ω = ( E min + E max ) will then blurthe pristine distinction between localised and delocalisedstates as a function of energy, required for optimal iden- tification of mobility edges. To circumvent this, it wouldbe preferable (sec. IV) to identify the band centre foreach disorder realisation from the centre of gravity Tr H of the eigenvalue spectrum, with energies referred relativeto that natural ‘origin’.Finally, we point out the obvious fact that a typical lo-cal coordination number for the FS lattice at non-zero fill-ing is its average, Z ∝ N (eq. 24), which thus grows un-boundedly in the thermodynamic limit. This is of courseradically different from the one-body case in any finitedimension d (although the limit of infinite coordinationnumber is familiar in the different context of dynamicalmean-field theory in d = ∞ ). A theory of localiza-tion in Fock-space must thus be able to explain how theoccurrence of a divergent coordination number is in ef-fect mitigated, such that an MBL transition exists in thethermodynamic limit N → ∞ . We will suggest one wayto do so in subsequent work. ACKNOWLEDGMENTS
Many helpful discussions with John Chalker, Sthi-tadhi Roy and Peter Wolynes are gratefully acknowl-edged. We also thank the EPSRC for support undergrant EP/L015722/1 for the TMCS Centre for DoctoralTraining, and grant EP/N01930X/1. D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann. Phys.(N.Y.) , 1126 (2006). R. Nandkishore and D. A. Huse, Annu. Rev. Condens.Matter Phys. , 15 (2015). E. Altman and R. Vosk, Annu. Rev. Condens. MatterPhys. , 383 (2015). D. E. Logan and P. G. Wolynes, J. Chem. Phys. , 4994(1990). D. M. Leitner, Adv. Phys. , 445 (2015). B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov,Phys. Rev. Lett. , 2803 (1997). V. Oganesyan and D. A. Huse, Phys. Rev. B , 155111(2007). D. E. Logan and S. Welsh, arXiv:1806.01688. Indeed for N e = 1 the states {| I (cid:105)} reduce to the orbitalbasis {| i (cid:105)} of the one-particle TBM. For 1BL the eigenvalue spectrum is not of course Gaussian, but the considerations here apply to the first and secondmoments of it, which is sufficient for our purposes. It is in fact well converged to the form shown after manyfewer disorder realisations than the 200 used to generatefig. 3. As expected from self-averaging, each realisation alsoyields essentially the same eigenvalue spectrum, modulomild differences attributable to the modest system size. D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B ,081103 (2015). S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H.Bardarson, Phys. Rev. Lett. , 046603 (2015). T. Devakul and R. R. P. Singh, Phys. Rev. Lett. ,187201 (2015). M. Serbyn, Z. Papi´c, and D. A. Abanin, Phys. Rev. X ,041047 (2015). A. Georges, G. Kotliar, W. Krauth, and M. J. Rosenberg,Rev. Mod. Phys.68